3.11.62 \(\int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac {\sqrt {1-x}}{3 \sqrt {x+1}}-\frac {\sqrt {1-x}}{3 (x+1)^{3/2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {45, 37} \begin {gather*} -\frac {\sqrt {1-x}}{3 \sqrt {x+1}}-\frac {\sqrt {1-x}}{3 (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*(1 + x)^(5/2)),x]

[Out]

-Sqrt[1 - x]/(3*(1 + x)^(3/2)) - Sqrt[1 - x]/(3*Sqrt[1 + x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx &=-\frac {\sqrt {1-x}}{3 (1+x)^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx\\ &=-\frac {\sqrt {1-x}}{3 (1+x)^{3/2}}-\frac {\sqrt {1-x}}{3 \sqrt {1+x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {1-x} (x+2)}{3 (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - x]*(1 + x)^(5/2)),x]

[Out]

-1/3*(Sqrt[1 - x]*(2 + x))/(1 + x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.05, size = 33, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {1-x} \left (\frac {1-x}{x+1}+3\right )}{6 \sqrt {x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[1 - x]*(1 + x)^(5/2)),x]

[Out]

-1/6*(Sqrt[1 - x]*(3 + (1 - x)/(1 + x)))/Sqrt[1 + x]

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fricas [A]  time = 1.35, size = 38, normalized size = 0.93 \begin {gather*} -\frac {2 \, x^{2} + {\left (x + 2\right )} \sqrt {x + 1} \sqrt {-x + 1} + 4 \, x + 2}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(2*x^2 + (x + 2)*sqrt(x + 1)*sqrt(-x + 1) + 4*x + 2)/(x^2 + 2*x + 1)

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giac [B]  time = 0.67, size = 89, normalized size = 2.17 \begin {gather*} \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{48 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{16 \, \sqrt {x + 1}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {9 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{48 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/2)/(1+x)^(5/2),x, algorithm="giac")

[Out]

1/48*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 3/16*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/48*(x + 1)^(3/2)
*(9*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3

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maple [A]  time = 0.00, size = 18, normalized size = 0.44 \begin {gather*} -\frac {\left (x +2\right ) \sqrt {-x +1}}{3 \left (x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+1)^(1/2)/(x+1)^(5/2),x)

[Out]

-1/3*(2+x)/(x+1)^(3/2)*(-x+1)^(1/2)

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maxima [A]  time = 2.93, size = 38, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

-1/3*sqrt(-x^2 + 1)/(x^2 + 2*x + 1) - 1/3*sqrt(-x^2 + 1)/(x + 1)

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mupad [B]  time = 0.31, size = 33, normalized size = 0.80 \begin {gather*} -\frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}}{\left (3\,x+3\right )\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(1/2)*(x + 1)^(5/2)),x)

[Out]

-(x*(1 - x)^(1/2) + 2*(1 - x)^(1/2))/((3*x + 3)*(x + 1)^(1/2))

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sympy [A]  time = 2.35, size = 65, normalized size = 1.59 \begin {gather*} \begin {cases} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(1/2)/(1+x)**(5/2),x)

[Out]

Piecewise((-sqrt(-1 + 2/(x + 1))/3 - sqrt(-1 + 2/(x + 1))/(3*(x + 1)), 2/Abs(x + 1) > 1), (-I*sqrt(1 - 2/(x +
1))/3 - I*sqrt(1 - 2/(x + 1))/(3*(x + 1)), True))

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